arXiv.orghttp://arxiv.org/icons/sfx.gifhttp://arxiv.org/
On the multiplicity and regularity index of toric curves. (arXiv:1909.05431v1 [math.AC])http://arxiv.org/abs/1909.05431
<p>In this note we revisit the problem of determining combinatorially the
multiplicity at the origin of a toric curve. In addition, we give the exact
value of the regularity index of that point for plane toric curves and
effective bounds for this number for arbitrary toric curves.
</p>
<a href="http://arxiv.org/find/math/1/au:+Duarte_D/0/1/0/all/0/1">Daniel Duarte</a>, <a href="http://arxiv.org/find/math/1/au:+Sandoval_A/0/1/0/all/0/1">Alondra Ram&#xed;rez Sandoval</a>Closure-interior duality over complete local rings. (arXiv:1909.05739v1 [math.AC])http://arxiv.org/abs/1909.05739
<p>We define a duality operation connecting closure operations, interior
operations, and test ideals, and describe how the duality acts on common
constructions such as trace, torsion, tight and integral closures, and
divisible submodules. This generalizes the relationship between tight closure
and tight interior given in [Epstein-Schwede 2014] and allows us to extend
commonly used results on tight closure test ideals to operations such as those
above.
</p>
<a href="http://arxiv.org/find/math/1/au:+Epstein_N/0/1/0/all/0/1">Neil Epstein</a>, <a href="http://arxiv.org/find/math/1/au:+G_R/0/1/0/all/0/1">Rebecca R.G</a>On the New Intersection Theorem for totally reflexive modules. (arXiv:1401.5716v4 [math.AC] UPDATED)http://arxiv.org/abs/1401.5716
<p>Let (R,m,k) be a local ring. We establish a totally reflexive analogue of the
New Intersection Theorem, provided for every totally reflexive R-module M,
there is a big Cohen-Macaulay R-module B_M such that the socle of B_M\otimes_RM
is zero. When R is a quasi-specialization of a G-regular local ring or when M
has complete intersection dimension zero, we show the existence of such a big
Cohen-Macaulay R-module. It is conjectured that if R admits a non-zero
Cohen-Macaulay module of finite Gorenstein dimension, then it is
Cohen-Macaulay. We prove this conjecture if either R is a quasi-specialization
of a G-regular local ring or a quasi-Buchsbaum local ring.
</p>
<a href="http://arxiv.org/find/math/1/au:+Divaani_Aazar_K/0/1/0/all/0/1">Kamran Divaani-Aazar</a>, <a href="http://arxiv.org/find/math/1/au:+Mashhad_F/0/1/0/all/0/1">Fatemeh Mohammadi Aghjeh Mashhad</a>, <a href="http://arxiv.org/find/math/1/au:+Tavanfar_E/0/1/0/all/0/1">Ehsan Tavanfar</a>, <a href="http://arxiv.org/find/math/1/au:+Tousi_M/0/1/0/all/0/1">Massoud Tousi</a>On the resolution of fan algebras of principal ideals over a Noetherian ring. (arXiv:1612.02939v5 [math.AC] UPDATED)http://arxiv.org/abs/1612.02939
<p>We construct explicitly a resolution of a fan algebra of principal ideals
over a Noetherian ring for the case when the fan is a proper rational cone in
the plane. Under some mild conditions on the initial data, we show that this
resolution is minimal.
</p>
<a href="http://arxiv.org/find/math/1/au:+Benitez_T/0/1/0/all/0/1">Teresa Cortadellas Benitez</a>, <a href="http://arxiv.org/find/math/1/au:+DAndrea_C/0/1/0/all/0/1">Carlos D&#x27;Andrea</a>, <a href="http://arxiv.org/find/math/1/au:+Enescu_F/0/1/0/all/0/1">Florian Enescu</a>Stanley-Reisner rings for symmetric simplicial complexes, G-semimatroids and Abelian arrangements. (arXiv:1804.07366v3 [math.CO] UPDATED)http://arxiv.org/abs/1804.07366
<p>We extend the notion of face rings of simplicial complexes and simplicial
posets to the case of finite-length (possibly infinite) simplicial posets with
a group action. The action on the complex induces an action on the face ring,
and we prove that the ring of invariants is isomorphic to the face ring of the
quotient simplicial poset under a mild condition on the group action. We also
identify a class of actions on simplicial complexes that preserve the
homotopical Cohen-Macaulay property under quotients. When the acted-upon poset
is the independence complex of a semimatroid, the $h$-polynomial of the ring of
invariants can be read off the Tutte polynomial of the associated group action.
Moreover, in this case an additional condition on the action ensures that the
quotient poset is Cohen-Macaulay in characteristic 0 and every characteristic
that does not divide an explicitly computable number. This implies the same
property for the associated Stanley-Reisner rings. In particular, this holds
for independence posets and rings associated to toric, elliptic and, more
generally, $(p,q)$-arrangements. As a byproduct, we prove that posets of
connected components (also known as posets of {layers}) of such arrangements
are Cohen-Macaulay with the same condition on the characteristic.
</p>
<a href="http://arxiv.org/find/math/1/au:+DAli_A/0/1/0/all/0/1">Alessio D&#x27;Al&#xec;</a>, <a href="http://arxiv.org/find/math/1/au:+Delucchi_E/0/1/0/all/0/1">Emanuele Delucchi</a>Generalized minimum distance functions and algebraic invariants of Geramita ideals. (arXiv:1812.06529v2 [math.AC] UPDATED)http://arxiv.org/abs/1812.06529
<p>Motivated by notions from coding theory, we study the generalized minimum
distance (GMD) function $\delta_I(d,r)$ of a graded ideal $I$ in a polynomial
ring over an arbitrary field using commutative algebraic methods. It is shown
that $\delta_I$ is non-decreasing as a function of $r$ and non-increasing as a
function of $d$. For vanishing ideals over finite fields, we show that
$\delta_I$ is strictly decreasing as a function of $d$ until it stabilizes. We
also study algebraic invariants of Geramita ideals. Those ideals are graded,
unmixed, $1$-dimensional and their associated primes are generated by linear
forms. We also examine GMD functions of complete intersections and show some
special cases of two conjectures of Toh\u{a}neanu--Van Tuyl and
Eisenbud-Green-Harris.
</p>
<a href="http://arxiv.org/find/math/1/au:+Cooper_S/0/1/0/all/0/1">Susan M. Cooper</a>, <a href="http://arxiv.org/find/math/1/au:+Seceleanu_A/0/1/0/all/0/1">Alexandra Seceleanu</a>, <a href="http://arxiv.org/find/math/1/au:+Tohaneanu_S/0/1/0/all/0/1">Stefan O. Tohaneanu</a>, <a href="http://arxiv.org/find/math/1/au:+Pinto_M/0/1/0/all/0/1">Maria Vaz Pinto</a>, <a href="http://arxiv.org/find/math/1/au:+Villarreal_R/0/1/0/all/0/1">Rafael H. Villarreal</a>